Properties

Degree 2
Conductor $ 2^{3} \cdot 7 \cdot 11 \cdot 13 $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.94·3-s + 1.48·5-s − 7-s + 0.765·9-s + 11-s + 13-s + 2.87·15-s − 5.79·17-s + 2.95·19-s − 1.94·21-s − 3.43·23-s − 2.80·25-s − 4.33·27-s − 2.27·29-s − 4.29·31-s + 1.94·33-s − 1.48·35-s − 7.63·37-s + 1.94·39-s − 3.26·41-s − 6.24·43-s + 1.13·45-s − 5.19·47-s + 49-s − 11.2·51-s + 14.1·53-s + 1.48·55-s + ⋯
L(s)  = 1  + 1.12·3-s + 0.663·5-s − 0.377·7-s + 0.255·9-s + 0.301·11-s + 0.277·13-s + 0.742·15-s − 1.40·17-s + 0.678·19-s − 0.423·21-s − 0.715·23-s − 0.560·25-s − 0.834·27-s − 0.423·29-s − 0.771·31-s + 0.337·33-s − 0.250·35-s − 1.25·37-s + 0.310·39-s − 0.510·41-s − 0.952·43-s + 0.169·45-s − 0.758·47-s + 0.142·49-s − 1.57·51-s + 1.94·53-s + 0.199·55-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 8008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 8008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(8008\)    =    \(2^{3} \cdot 7 \cdot 11 \cdot 13\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{8008} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 8008,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;7,\;11,\;13\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;7,\;11,\;13\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 \)
7 \( 1 + T \)
11 \( 1 - T \)
13 \( 1 - T \)
good3 \( 1 - 1.94T + 3T^{2} \)
5 \( 1 - 1.48T + 5T^{2} \)
17 \( 1 + 5.79T + 17T^{2} \)
19 \( 1 - 2.95T + 19T^{2} \)
23 \( 1 + 3.43T + 23T^{2} \)
29 \( 1 + 2.27T + 29T^{2} \)
31 \( 1 + 4.29T + 31T^{2} \)
37 \( 1 + 7.63T + 37T^{2} \)
41 \( 1 + 3.26T + 41T^{2} \)
43 \( 1 + 6.24T + 43T^{2} \)
47 \( 1 + 5.19T + 47T^{2} \)
53 \( 1 - 14.1T + 53T^{2} \)
59 \( 1 + 0.168T + 59T^{2} \)
61 \( 1 + 1.15T + 61T^{2} \)
67 \( 1 + 5.79T + 67T^{2} \)
71 \( 1 + 13.7T + 71T^{2} \)
73 \( 1 + 9.12T + 73T^{2} \)
79 \( 1 + 3.90T + 79T^{2} \)
83 \( 1 + 1.99T + 83T^{2} \)
89 \( 1 - 1.67T + 89T^{2} \)
97 \( 1 - 15.7T + 97T^{2} \)
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\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−7.46013011026792242994359940734, −6.93569101918329020162900867671, −6.06898939821666319594285871390, −5.54790194162172823701305838181, −4.51646660859508527641408036367, −3.68883759581093773933177044480, −3.13713166117487649157188235924, −2.15350763867920484572756021275, −1.69431119173604730834224934210, 0, 1.69431119173604730834224934210, 2.15350763867920484572756021275, 3.13713166117487649157188235924, 3.68883759581093773933177044480, 4.51646660859508527641408036367, 5.54790194162172823701305838181, 6.06898939821666319594285871390, 6.93569101918329020162900867671, 7.46013011026792242994359940734

Graph of the $Z$-function along the critical line